In fact there was never any time in Western intellectual history when this was true. There may have been a few individuals here or there who thought that the earth is flat, but the vast majority of scientists, philosophers, theologians, and other interested parties all knew that the earth is roughly spherical. Why do so many students think otherwise? One reason might be because the myth has been repeated by some people who ought to know better. For example, Daniel Boorstin repeats it in his 1983 book The Discoverers, and Boise Penrose mentions it in his 1955 book Travel and Discovery in the Renaissance. So where did this idea come from in the first place? Lesley Cormack has traced it back to a biography of Christopher Columbus written by Washingon Irving (of "Rip Van Winkle" fame). According to Cormack, Irving was trying to portray Columbus as a landmark figure, rather like Copernicus, who by virtue of his voyage of discovery put to rest one more vestige of the so-called "dark ages". In short, it was a piece of propaganda.
Of course, one cannot expect your average undergraduate to be so well-versed in these matters, and so it is perhaps unsurprising that they simply accept whatever they hear along these lines. Unsurprising, but still disappointing, since so many undergraduates these days like to portray themselves as skeptical inquirers after the truth who are not willing to blindly accept everything they are told. It turns out they are just like the rest of us in that regard.
So what sort of remedy is there for this kind of situation? I would like to suggest that one step in the right direction would be to re-integrate the study of the sciences with the study of intellectual history. Many of the best contemporary science writers are people who not only know a lot about science, they know a lot about the history and philosophy of science as well. More importantly, they are sensitive readers who are able to ferret out important details that can aid in the formation of more subtle interpretations of that intellectual history.
There is an interesting illustrative example early on in Bede's Ecclesiastical History of the English People. In I.3 he writes, in part:
Ireland is far more favored than Britain by latitude, and by its mild and healthy climate. Snow rarely lies longer than three days, so that there is no need to store hay in summer for winter use or to build stables for beasts. There are no reptiles, and no snake can enter there; for although often brought over from Britain, as soon as the ship nears land, they breathe the scent of its air, and die. In fact, almost everything in this isle confers immunity to poison, and I have seen that folk suffering from snake-bite have drunk water in which scrapings from the leaves of books from Ireland have been steeped, and that this remedy checked the spreading poison and reduced the swelling.The last sentence of this report, in particular, is apt to make one roll one's eyes at the credulity of an earlier age, but H. Mayr-Harting, in his book The Coming of Christianity, has argued at some length that in fact this passage is not meant to be taken seriously, but is rather a parody, by Bede, of the sort of pseudo-scientific writing common in his day. Seeing this requires knowing, first of all, that Bede was a rather witty writer not above making jokes or parodies, and also knowing about the pseudo-scientific literature that he would have had access to. Again, this is not the sort of thing that we can expect just anybody to know off-hand, but clearly knowledge of this sort of thing would make a difference in how one interprets this passage of Bede.
I suspect that there are many such misapprehensions abroad. I think that much contemporary discussion of the conflict between Galileo and the Church, for example, suffers from a lack of full information on both sides, and that even when it comes to such obvious non-problems as the "conflict" between evolution and intelligent design, or anthropogenic climate change, the clearer heads are the ones who understand fully what sorts of issues are at stake on all sides and who, more importantly, are careful enough to know the difference between principled arguments and ideological scrums. This requires wide reading, and a careful assessment of merits of the arguments on all sides. The intellectual acumen required to accomplish this ranges more widely, I am afraid, than the typical training either in the sciences or the humanities can give. A broader education is required, and those who are fluent in both will have the best chance of making an accurate assessment of the situation.